# Scalar Multiplication of Matrices

In matrix algebra, a real number is called a scalar .

The scalar product of a real number, $r$ , and a matrix $A$ is the matrix $rA$ .  Each element of matrix $rA$ is $r$ times its corresponding element in $A$ .

Given scalar $r$ and matrix $A=\left[\begin{array}{cc}{a}_{11}& {a}_{12}\\ {a}_{21}& {a}_{22}\end{array}\right],\text{\hspace{0.17em}}\text{\hspace{0.17em}}rA=\left[\begin{array}{cc}r{a}_{11}& r{a}_{12}\\ r{a}_{21}& r{a}_{22}\end{array}\right]$ .

Example 1:

Let $A=\left[\begin{array}{cc}\hfill 2& \hfill 1\\ \hfill 3& \hfill -2\end{array}\right]$ , find $4A$ .

$4A=4\left[\begin{array}{cc}\hfill 2& \hfill 1\\ \hfill 3& \hfill -2\end{array}\right]=\left[\begin{array}{cc}4\cdot 2& 4\cdot 1\\ 4\cdot 3& 4\cdot \left(-2\right)\end{array}\right]=\left[\begin{array}{cc}\hfill 8& \hfill 4\\ \hfill 12& \hfill -8\end{array}\right]$

## Properties of Scalar Multiplication:

Let $A$ and $B$ be $m×n$ matrices.  Let ${O}_{m×n}$ be the $m×n$ zero matrix and let $p$ and $q$ be scalars.

 Properties of Scalar Multiplication Associative Property $p\left(qA\right)=\left(pq\right)A$ Closure Property $pA$ is an $m×n$ matrix. Commutative Property $pA=Ap$ Distributive Property $\begin{array}{l}\hfill \left(p+q\right)A=pA+qA\hfill \\ \hfill p\left(A+B\right)=pA+pB\hfill \end{array}$ Identity Property $1\cdot A=A$ Multiplicative Property of $-1$ $\left(-1\right)A=-A$ Multiplicative Property of $0$ $0\cdot A={O}_{m×n}$